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Commutation Relations

The commutation relations between operators, such as [\hat{x}, \hat{p}] = iħ, are fundamental to the theory, dictating the uncertainty principle.
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The statement of the theorem

Let H\mathcal{H} be a separable Hilbert space, and let X^\hat{X} and P^\hat{P} be self-adjoint operators defined on H\mathcal{H} representing position and momentum, respectively. The commutation relation is defined by the commutator [X^,P^]≡X^P^−P^X^[\hat{X}, \hat{P}] \equiv \hat{X}\hat{P} - \hat{P}\hat{X}. The canonical commutation relation (CCR) asserts that for the standard basis representation, the commutator yields: [X^,P^]=iℏI^[\hat{X}, \hat{P}] = i\hbar \hat{I} where ℏ\hbar is the reduced Planck constant and I^\hat{I} is the identity operator on H\mathcal{H}. Furthermore, for any two observables A^\hat{A} and B^\hat{B}, the expectation value of the commutator satisfies ⟨ψ∣[A^,B^]∣ψ⟩=iℏ⟨ψ∣∂A^∂q∂B^∂p−∂B^∂q∂A^∂p∣ψ⟩\langle \psi | [\hat{A}, \hat{B}] | \psi \rangle = i\hbar \langle \psi | \frac{\partial \hat{A}}{\partial q} \frac{\partial \hat{B}}{\partial p} - \frac{\partial \hat{B}}{\partial q} \frac{\partial \hat{A}}{\partial p} | \psi \rangle. This structure dictates the uncertainty principle ΔAΔB≥12∣⟨[A^,B^]⟩∣\Delta A \Delta B \ge \frac{1}{2} |\langle [\hat{A}, \hat{B}] \rangle|.